------------- Miles Reid, in his _Undergraduate Algebraic Geometry_ tells with evident horror that: I actually know of a thesis on the arithmetic of cubic surfaces that was initially not considered because 'the natural context for the construction is over a general locally Noetherian ringed topos'. This is not a joke. p.116 Now, I know little about the arithmetic of surfaces. I do know that classical cubic surfaces are pretty well understood at least in some ways. Are there substantial open questions about their arithmetic? Basically my question is just what is the point here? Is Reid objecting that arithmetic of cubic surfaces is already so hard in the classical case that it is crazy to look at a more general one? Or is he saying it is so easy that it is crazy to make it hard by bringing in toposes? Any help on this will be much appreciated. Colin McLarty +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
Well, there is a lot of problems in the arithmetic of cubical surfaces. There is a book by Yu.I.Manin called "Cubical surfaces" which was written in sixties and as far as I know most of the conjectures are still open (see also recent reviews by Manin & Co.. on the arithmetic of rational varities). It is very nice and hard part of the arithmetic algebraic geometry which is in a way a generalization of the classical arithmetic of elliptic curves. I really doubt that toposes (especially ringed toposes...) can be of much help here mostly because the theory of toposes is much more elementary thing than all these arithmetical problems. Vladimir Voevodsky. +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
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